The Rotation of Logarithmic-Spiral-Shaped Gears Adam Jaffe March 8, 2013
Abstract
This paper dissects the rotation of spiral-shaped gears, their interesting properties, their general rates of change, and their unique use in mechanical engineering.
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The System
Two identical gears are placed tangent to one another, their centers separated by a constant distance. Each gear has a spiral curve, and for this investigation, we will assume the spiral to be logarithmic. This is an example of a gear with an uneven uneven surface, surface, often often called called a “ca “cam.” m.” An uneven uneven surface surface gives gives a gea gearr interesting properties, so cams are very common in mechanical engineering. This will be investigated more thoroughly later. A visualization of the system can be seen here. (Although the camerawork is shaky, the system still behaves normally.)
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Repr Repres esen enti ting ng the the Gea Gears rs
Before we can investigate the movement of this system, we need a compact, mathematical way to represent each gear. Because of the nature of gears and cams, we choose to represent this specific gear with a logarithmic spiral, of the polar equation: r(θ) = hθ/2π (1) This polar equation describes a cam of a logarithmic spiral containing the points r(0) = 1 and r (2π (2π ) = h. Note Note that we restrict restrict rotation rotation to that of 2π 2π radians to ensure that the system can be represented physically with some key properties. 1
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Gear Rotation
When two gears rotate, their tangent surface length displacement must be equal. equal. (This (This is why why rotating rotating one large gear and one small gear results results in a disparity disparity of angular angular velocity velocity.) .) Naturally Naturally,, we now want to represen representt tangent tangent surface length displacement as a function of the rotation of each cam. The general length of a polar function is given by: b
r2 + (dr/dθ (dr/dθ))2 dθ
(2)
a
For our previously determined polar equation, the total length is simply:
2π
hθ/π
+
0
hθ/2π ln h 2π
2
dθ
(3)
However, we don’t need the total length of the cam, but actually the tangent gent surface surface length displacement displacement for an arbitrary arbitrary amount of rotation. rotation. Since two tangent gears rotate in opposite directions, the integrals of displacement for both gears will be similar, but slightly different (Here we use α and β for the rotations of the two gears respectively):
α
hθ/π
+
hθ/2π ln h 2π
+
hθ/2π ln h 2π
0
2π
hθ/π
β
2
dθ
(4)
dθ
(5)
2
It should now be very clear that since the two tangent gears rotate in opposit oppositee direct directions ions,, they they must must sum to the total surfac surfacee length length of the gear when their amounts of rotation are equal. Evaluating the above integrals yields: (hα/2 − 1)
4 1 + 2 2 ln h π
(6)
4 1 + (7) ln2 h π 2 Finally, we want to relate the rotations of the gears, α and β , in one concise equation. We know that tangent surface length displacement must be constant between the two gears, so: (hπ − hβ/ 2 )
2
4 1 4 1 π β/ 2 − (hα/2 − 1) + = ( h h ) + (8) ln2 h π2 ln2 h π 2 The radical factor is the same in both expressions, so the relationship is simply: hα/2 − 1 = hπ − hβ/ 2
(9)
Or perhaps, the more convenient version: hα/2 + hβ/ 2 = hπ + 1
4
(10)
Angu Angula lar r Veloci elocitty
We have now established a relationship between the rotation of each gear. The natural follow-up question is of course that of the angular velocity of each gear. Simple implicit different differentiation iation yields (remember that h is a constant!): dα α/2 dβ β/ 2 h + h =0 dt dt
5
(11)
Uses Us es in Mec Mechani hanica call Eng Engin inee eeri ring ng
This system of gears has many interesting properties that are especially useful in mechanical mechanical engineering engineering.. The relationship relationship we have determined determined can be graphed as a function, but there is one important restriction we must remember to enforc enforce. e. We initiall initially y limite limited d the gears’ gears’ rotation rotation from 0 to 2π so that it would would be physically physically represent representable able without without overlap overlapping ping itself. Howev However, er, this limitation also makes the gears periodic, of period 2π 2π . When we graph this function with the restriction, it is visible that every period period ends ends with a sharp sharp decline. decline. This This is an importa important nt quality quality because because this system system of gears can be used to create create a periodic p eriodic null null in rotation. As long as one gear can maintain a constant angular velocity (They cannot both be constant!), the output of the other gear will have a periodic null. This mechanical device is importan importantt to creatin creatingg this this specific specific signal, signal, used used com common monly ly in clocks clocks and other repetitive repetitive machiner machinery y. In the previous .gif, it is evident evident that when one gear moves with constant angular velocity (Although it may not seem so, the bottom gear has constant rotation.) the other gear rotates in a pattern dictated by our previously discovered equations.
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